Abstract
We will explain Arnold's 2-dimensional (shortly, 2D) projective geometry (Arnold, 2005) by means of lattice theory. It will be shown that the projection of the set of nontrivial triangular -matrices is the pencil of tangent lines of a quadratic curve on Arnold's projective plane.
1. Introduction
We briefly describe Arnold's projective geometry [1]. We recall the ordinary projective plane over the real field. We replace to the set of the quadratic functions on the canonical symplectic plane . The quadratic functions form into a Lie subalgebra of the canonical Poisson algebra on the symplectic plane. This Lie algebra is isomorphic to that is, Arnold introduced a projective plane . By the projection, a (nontrivial) quadratic function corresponds to a point on the projective plane, and the Killing form on corresponds to a quadratic curve on the projective plane, because it is a symmetric bilinear form. Since the Killing form is nondegenerate, the associated curve defines a duality (so-called polar system) between the projective lines and the projective points. Arnold showed that the Poisson bracket corresponds to the pole point of the projective line through and . As an application, it was shown that given a good triangle composed of three points , the three altitudes intersect the same point (altitude theorem). Interestingly, the altitude theorem is shown by the Jacobi identity (see Figure 1).
The monomials of double brackets correspond to the three bold lines, via the line-point duality. By the Jacobi identity, the three monomials are linearly dependent. This implies that the three lines intersect the same point.
We suppose that 2D projective geometry is encoded in the Lie algebra. However some information on the Lie algebra is lost in the process of constructing projective geometry; besides, it is not clear why it has to happen, conceptually. So we will reformulate the Arnold construction by means of lattice theory. Since the lattice is an algebra, the problem becomes more clear. We will prove, when the characteristic of the ground field is not , that each 3D simple Lie algebra admits a modular lattice structure. This proposition explains why 2D projective geometry can be encoded in .
It is crucial to consider the PlΓΌcker embedding for the algebraic Arnold construction. When is 3D and simple, the Lie algebra multiplication , is an isomorphism, which induces an isomorphism . We will show that the line-point duality on is equivalent with the isomorphism , up to the PlΓΌcker embedding.
As an application we study a projective geometry of triangular -matrices, when . We will prove that the projection of the set of nontrivial triangular -matrices is equivalent to the pencil of tangent lines on the quadratic curve made from the Killing form. This proposition is equivalently translated as follows: the classical Yang-Baxter equation on is equivalent to the quadratic curve on the projective plane.
2. Algebraic Arnold Construction
2.1. Subspace Lattices
A lattice is, by definition, a set equipped with two commutative associative multiplications, and , satisfying the following two identities: A lattice has a canonical order defined by A lattice is called a modular lattice when it satisfies the inequality (modular rule) The notion of lattice morphism is defined by the usual manner.
Example 2.1 (subspace lattices). Let be a vector space. Consider the set of subspaces of : . Define two natural multiplications (cup- and cap-products) on by where . Then becomes a modular lattice. The induced order is the natural inclusion relation . Given a linear injection , an associated lattice morphism is naturally defined by .
The subspace lattices are complementary; that is, the zero space is the unit element with respect to and the total space is . If is split for each , then there exists a cosubspace satisfying and . The subspace (resp., ) is called a complement1 of (resp., ). Such a lattice is called a complemented lattice. If is finite dimensional, then the subspace lattice is a complemented-modular-lattice. A projective geometry is axiomatically defined as a complemented-modular-lattice satisfying some additional properties.
Definition 2.2. When is -dimensional, the subspace lattice is called an -dimensional projective geometry over .
The 1D subspaces are regarded as projective points, 2D subspaces are projective lines and so on. The zero space is regarded as the empty set. For instance, given two 2D subspaces and , the intersection is the common point of two projective lines (if it exists).
2.2. Lie Algebra Construction of Projective Plane
Let be a 3D -Lie algebra with a nondegenerate symmetric invariant pairing , where is the ground field . We assume that , or equivalently, the Lie bracket is an isomorphism from to . This assumption is needed in order to construct projective geometry. Such a Lie algebra is simple. In particular, when is a closed field, is isomorphic to .
We denote by the 1D subspace generated by and denote by the 2D subspace generated by . We define new cup-and cap-products and on .
Definition 2.3 (Arnold products). (i) , , where means the orthogonal space with respect to the invariant pairing .
(ii) ,
(iii) and , in all other cases.
Proposition 2.4. The set with Arnold products becomes a lattice and it is the same as the classical subspace lattice.
Proof. By the invariancy of the pairing, we have . This gives Hence we have . We prove that . Since , we have , and . Thus we obtain which gives .
In the following, we omit the βprimeβ on the Arnold products.
Remark 2.5 (Lie algebra identities versus lattice identities). We put . Then the Arnold products are coherent with the Lie bracket, namely, Tomihisa [2] discovered an interesting identity on : where are fixed. The Tomihisa identity induces a lattice identity where , and . In a study by Aicardi in [3], it was shown that the projection of Tomihisa identity is equivalent to the Pappus theorem. (The author also proved this proposition around winter 2007.) We leave it to the reader to write down the lattice identity associated with the Jacobi identity.
Given a coordinate on the projective plane, the symmetric pairing is regarded as a defining equation of a nondegenerate quadratic curve (so-called polar system): We give an example of the quadratic curve made from the pairing.
Example 2.6 (see, [1, 2]). We assume that generated by the standard basis satisfying the following relations: The symmetric pairing is equivalent with the Killing form. We define the scale of the form by and all others zero. We set a point , . It is on the quadratic curve, that is, if and only if , and this condition is equivalent with , via the coordinate transformation Hence the quadratic curve is regarded as a circle on the projective plane with coordinate .
Remark 2.7. The pairing induces a metric on . It is well known that the inside of the circle (so-called timelike subspace) is a hyperbolic plane. The altitude theorem is strictly a theorem on the hyperbolic plane because a metric is needed to define the notion of altitude.
Given a point , a line is defined by This line is called the polar line of the point; conversely the point is called the pole of the line. Namely, the orthogonal space is the pole of the line .
Corollary 2.8 (see [1]). Given a line , is the pole of the line.
Figure 2 is depicting the duality defined by an ellipse.
The line is the polar line of the point which is inside an ellipse. The line and point are connected by the tangent lines and chords of the ellipse. This figure has been drawn in a book by Kawada in [4].
We did not use Jacobi identity in this section. So we will discuss the Jacobi identity on the Lie algebra in the next section.
2.3. Jacobi Identity
Proposition 2.9. Let be a 3D -vector space equipped with a skew-symmetric bracket product and a nondegenerate symmetric invariant pairing . Then the bracket satisfies the Jacobi identity.
Proof. When are linearly dependent, Jacobi identity holds. So we assume that is linear independent.Case 1. By the invariancy of the pairing, we have We assume that . Then we obtain Since the pairing is nondegenerate, we have , which gives the Jacobi identity .Case 2. Assume that . We already saw that . We show that . Since , one can put , for some . Then we have Therefore is orthogonal with the independent two elements and . (If and are linearly dependent, then .) The monomial is also orthogonal with and . Thus we obtain We can assume that or , because the pairing is nondegenerate. We assume that without loss of generality. Then we obtain , because
The proposition above indicates that the Jacobi identity is a priori invested in the projective plane.
2.4. Duality Principle
Let be the Lie algebra structure on ; that is, , . Since is an isomorphism, it induces a lattice isomorphism Let be the set of all lines in . One can define an injection as
This mapping is called a PlΓΌcker embedding.
Proposition 2.10. The diagram below is commutative:
(2.22)
where is the set of points on the projective plane and is the duality correspondence.
Namely, the Lie algebra multiplication is equivalent to the duality principle.
3. -Matrices
In this section, we assume that , . We consider a graded commutative algebra A graded Poisson bracket with degree is uniquely defined on by the axioms of graded Poisson algebra and the condition This bracket is called a Schouten-Nijenhuis bracket. Let be a 2 tensor in . The Maurer-Cartan (MC) equation is called a classical Yang-Baxter equation and the solution is called a classical triangular -matrix. For instance, is a solution of the MC-equation.
Remark 3.1. When , there is no nontrivial triangular -matrix.
Proposition 3.2. The set of points associated with nontrivial triangular -matrices bijectively corresponds to the pencil of tangent lines of the quadratic curve made from the symmetric pairing, or, equivalently, bijectively corresponds to the quadratic curve.
Proof. Since , one can write for some . By the biderivation property of the Schouten-Nijenhuis bracket, we have
Hence if and only if is linearly dependent on and .Lemma 3.3. The set of lines including the pole corresponds to , via the PlΓΌcker embedding.
If , then the pairing vanishes, because . Hence is on the quadratic curve. It is easy to check that is not cross over the curve. Hence the line is tangent to the curve at .
Lemma 3.4. A line is tangent to the quadratic curve if and only if the pole is on the line, and the tangent point is the pole of the line (see Figure 3). Therefore is identified with the pencil of tangent lines of the quadratic curve. The proof of the proposition is completed.
Acknowledgment
The author would like to express his thanks to Professor Akira Yoshioka and Dr. Toshio Tomihisa for their helpful comments.
Endnotes
- The complement is not unique in general.